Wave-breaking phenomena and persistence properties for a nonlinear dissipative Camassa–Holm equation

Abstract

In present paper, we mainly consider the wave-breaking phenomena and persistence properties of a nonlinear dissipative Camassa–Holm equation. The precise blow-up criteria of the Cauchy problem are established and wave-breaking phenomenon is investigated based on the method of characteristics and the Riccati-type differential inequality. Due to the presence of high order nonlinear term, the conservation law $E={\int }_{\mathbb{R}}{u}^2+u_x^2\mathrm{d}x$ is losed. This difficulty has been addressed by establishing the energy inequality. Finally, we establish the persistence properties and some unique continuation properties of the solutions in the weighted $L^p$-spaces $L_{\varphi }^p\left(\mathbb{R}\right):=L^p(\mathbb{R},{\varphi }^p\mathrm{d}x)$.

Keywords:

• Generalized Camassa–Holm equation
• high order nonlinear term
• wave breaking
• persistence properties

AMS:

• 35B30
• 35G25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is partially supported by the National Natural Science Foundation of China [No. 11701068].